Solve Problems With No Time Step Restriction Using the Implicit Finite Difference Method

Key Takeaways

• The implicit finite difference method is used for problem-solving when the output expression at a forward time step depends on itself. 

• There will be more than one unknown in the implicit finite difference equation.

• The implicit finite difference method is generally used to solve problems involving no restrictions on the time step. 

Numerical methods are employed to solve partial differential equations

To solve partial differential equations, numerical methods are often employed. Numerical methods based on difference techniques such as pseudospectral (PS), finite element (FM), and finite difference (FD) are used to solve heat conduction problems, fluid flow problems, and diffusion problems. The finite difference method can be explicit or implicit, depending on the type of equations developed for the given system. In the implicit finite difference method, there is no need for casual recursive computation, as the function depends on itself.

Let’s learn more about the implicit finite difference method.

Analytical Methods of Solving Partial Differential Equations

Numerical models of processes or systems are represented using partial differential equations in engineering and science. Mathematical models based on partial differential equations are solved to obtain the problem solution. The analytical method of problem-solving is applicable to partial differential equations only if the system has simple boundaries. However, most practical problems involve complicated boundary conditions or irregular boundaries. In systems modeled as difficult boundary value problems, the analytical method does not work. For such complex mathematical models, problem-solving involves the use of numerical methods.

Finite Difference Method

Difference techniques include pseudospectral (PS), finite element (FM), and finite difference (FD) methods. Among these numerical methods, the finite difference method is of great importance, as it requires minimum memory and computation time. Moreover, it involves a straightforward implementation with less complexity compared to other numerical techniques.

Apart from the conventional finite difference method, there are several variants available. The various finite difference variants are developed with the intent to increase the accuracy, efficiency, and stability of the finite difference method in numerical modeling.

Finite Difference Method Variants

When solving partial differential equations using analytical methods, solutions are closed-form expressions that convey the variation of the dependent variables in the problem domain. However, a finite difference method-based solution gives the values of the variables at discrete points in the domain. The discrete points are typically called grid points.

Grid Points

In the conventional finite difference method or scheme, the number of grid points is fixed. The conventional finite difference method requires large memory and computational time. In order to reduce the memory requirement and computational time, variable grid schemes are employed. Further, a reduction in computational costs can be achieved. The introduction of temporal sampling in different parts of the numerical grid not only minimizes the computation time but also optimizes the mesh size.

Based on the nature of the equations formulated for the problem domain, the finite difference method is classified into explicit and implicit finite difference methods. 

Distinguishing Explicit and Implicit Finite Difference Methods 

Among the variants of finite difference methods, explicit and implicit finite difference methods are invariably used. 

Explicit Finite Difference Method

While solving an equation, if the dependent variable at a time level is obtained directly from known values, it forms the explicit finite difference method. Consider the equation:

In this equation, the value of y at time point (n+1) is dependent on variable x at time n and function of y at the time step n. This equation implies that the computation is carried out to obtain the value forward in time using quantities from previous time steps. The finite difference scheme of such type is said to be explicit.

However, in certain expressions, the output at a forward time step depends on itself. The implicit finite difference method is used in such problem-solving. 

Implicit Finite Difference Method

If the unknown at a future time level is expressed in terms of the variables at that time level and variables at past, present, and future times, it forms the implicit finite difference method.

Consider the equation: 

Here, the y at (n+1)^th time step depends on the x value at n^th time step and the function of f(y) at (n+1)th instant. There is no explicit relationship present in the equation. This calls for an implicit finite difference method.

Problem Solving Using the Implicit Finite Difference Method

The implicit finite difference method is generally used for solving problems involving no restrictions on the time step. This method is used to solve heat conduction equations, steady and unsteady inviscid and viscous compressible flow, diffusion equations, electromagnetic problems, and computing vortex wakes.

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